Let's break down the assignment step by step.
Question 1: Write the values given in the scenario as a sequence.
The salaries over the four years can be written as a sequence:
Sequence:
48000, 50400, 52920, 55566
Question 2: Select the appropriate answer.
This is an example of a geometric sequence because Willow's salary increases by a constant percentage each year.
Question 3: Create a function that models this sequence/scenario.
To model the sequence using a function, we can determine the common ratio (the percentage increase) first. We can calculate the percentage increase year over year.
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Calculate the percentage increase between the years:
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From $48,000 to $50,400: \[ \text{Percentage Increase} = \left(\frac{50400 - 48000}{48000}\right) \times 100 = \left(\frac{2400}{48000}\right) \times 100 = 5% \]
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From $50,400 to $52,920: \[ \text{Percentage Increase} = \left(\frac{52920 - 50400}{50400}\right) \times 100 = \left(\frac{2520}{50400}\right) \times 100 = 5% \]
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From $52,920 to $55,566: \[ \text{Percentage Increase} = \left(\frac{55566 - 52920}{52920}\right) \times 100 = \left(\frac{2646}{52920}\right) \times 100 = 5% \]
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The percentage increase each year is consistently 5%.
Now that we know the percentage increase is 5%, we can create the function.
The formula for a geometric sequence can be given by: \[ S(n) = a \cdot r^{(n - 1)} \] where:
- \( S(n) \) is the salary in the nth year,
- \( a \) is the initial salary ($48,000),
- \( r \) is the common ratio (1 + percentage increase as a decimal; for 5%, it is 1.05),
- \( n \) is the year number.
Thus, the function modeling Willow's salary sequence would be:
Function:
\[
S(n) = 48000 \cdot (1.05)^{(n - 1)}
\]
Summary of Answers:
- Sequence: 48000, 50400, 52920, 55566
- Type of sequence: Geometric sequence
- Function: S(n) = 48000 * (1.05)^{(n - 1)}
- Percentage increase: 5%